Theorem ineccnvmo 39209

Description: Equivalence of a double restricted universal quantification and a restricted "at most one" inside a universal quantification. (Contributed by Peter Mazsa, 2-Sep-2021.)

Assertion

Ref	Expression
ineccnvmo	⊢ (∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝑦 = 𝑧 ∨ ([𝑦]◡𝐹 ∩ [𝑧]◡𝐹) = ∅) ↔ ∀𝑥∃*𝑦 ∈ 𝐵 𝑥𝐹𝑦)

Distinct variable groups:   𝑥,𝐵,𝑦,𝑧   𝑥,𝐹,𝑦,𝑧

Proof of Theorem ineccnvmo

Step	Hyp	Ref	Expression
1		relcnv 6094	. . 3 ⊢ Rel ◡𝐹
2		id 23	. . . 4 ⊢ (𝑦 = 𝑧 → 𝑦 = 𝑧)
3	2	inecmo 39207	. . 3 ⊢ (Rel ◡𝐹 → (∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝑦 = 𝑧 ∨ ([𝑦]◡𝐹 ∩ [𝑧]◡𝐹) = ∅) ↔ ∀𝑥∃*𝑦 ∈ 𝐵 𝑦◡𝐹𝑥))
4	1, 3	ax-mp 5	. 2 ⊢ (∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝑦 = 𝑧 ∨ ([𝑦]◡𝐹 ∩ [𝑧]◡𝐹) = ∅) ↔ ∀𝑥∃*𝑦 ∈ 𝐵 𝑦◡𝐹𝑥)
5		brcnvg 5853	. . . . 5 ⊢ ((𝑦 ∈ V ∧ 𝑥 ∈ V) → (𝑦◡𝐹𝑥 ↔ 𝑥𝐹𝑦))
6	5	el2v 3457	. . . 4 ⊢ (𝑦◡𝐹𝑥 ↔ 𝑥𝐹𝑦)
7	6	rmobii 3373	. . 3 ⊢ (∃*𝑦 ∈ 𝐵 𝑦◡𝐹𝑥 ↔ ∃*𝑦 ∈ 𝐵 𝑥𝐹𝑦)
8	7	albii 1852	. 2 ⊢ (∀𝑥∃*𝑦 ∈ 𝐵 𝑦◡𝐹𝑥 ↔ ∀𝑥∃*𝑦 ∈ 𝐵 𝑥𝐹𝑦)
9	4, 8	bitri 278	1 ⊢ (∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝑦 = 𝑧 ∨ ([𝑦]◡𝐹 ∩ [𝑧]◡𝐹) = ∅) ↔ ∀𝑥∃*𝑦 ∈ 𝐵 𝑥𝐹𝑦)

This proof depends on syntax axioms:   ↔ wb 209   ∨ wo 861  ∀wal 1568   = wceq 1570  ∀wral 3076  ∃*wrmo 3364  Vcvv 3450   ∩ cin 3897  ∅c0 4278   class class class wbr 5102  ^◡ccnv 5646  Rel wrel 5652  [cec 8693

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-11 2194  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-mo 2564  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rmo 3365  df-rab 3413  df-v 3452  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-br 5103  df-opab 5167  df-xp 5653  df-rel 5654  df-cnv 5655  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-ec 8697

This theorem is used by:  ineccnvmo2  39220